On a Coupled System of Random and Stochastic Nonlinear Differential Equations with Coupled Nonlocal Random and Stochastic Nonlinear Integral Conditions

: It is well known that Stochastic equations had many useful applications in describing numerous events and problems of real world, and the nonlocal integral condition is important in physics, ﬁnance and engineering. Here we are concerned with two problems of a coupled system of random and stochastic nonlinear differential equations with two coupled systems of nonlinear nonlocal random and stochastic integral conditions. The existence of solutions will be studied. The sufﬁcient condition for the uniqueness of the solution will be given. The continuous dependence of the unique solution on the nonlocal conditions will be proved.


Introduction
Let (Ω, , P) be a fixed probability space, where Ω is a sample space, is a σ-algebra and P is a probability measure.
The aim of this article is to extend the results of A.M.A. El-Sayed [1,2] on the stochastic fractional calculus operators defined on C([0, T], L 2 (Ω)) and the solution of stochastic differential equations subject to nonlocal integral conditions which have been considered in [3,4].
Moreover, we motivate the coupled system of integral equations in reflexive Banach space by A.M.A. El-Sayed and H.H.G.Hashem [5] to the coupled systems with random memory on the space of all second order stochastic process.
The continuous dependence of a unique solution has been studied on the random initial data and the random function which ensures the stability of the solution.

Solutions of the Problem (1)-(3)
Define the mapping (F(x, y))(t) = (F 1 y, F 2 x)(t), t ∈ [0, T] where (F 1 y)(t), (F 2 x)(t) are given by the following stochastic integral equations Consider the set Q such that Now, we have the following two lemmas Proof. Let y ∈ Q, y(t) 2 ≤ r 1 , then This proves that F : Q → Q and the class of functions {F(x, y)} is uniformly bounded on Q.

Lemma 3. The class of functions {F(x, y)} is equicontinuous on Q.
This proves the equicontinuity of the class {F 1 y} and This proves the equicontinuity of the class {F 1 x}. Now then from (14) and (15), we can deduce the equicontinuity of the class {F(x, y)} on Q.

Existence Theorem
Now, we have the following existence theorem Theorem 1. Let the Assumptions 1-5 be satisfied, then there exists at least one solution (x, y) ∈ X of the problem (1)-(3).
Proof. Firstly, from the results of Lemmas 2 and 3 and Arzela-Ascoli Theorem [9] we deduce that the closure of FQ is a compact subset. Let (x n , y n ) ∈ Q be such that where L.i.m denotes the limit in the mean square sense of the continuous second order process ( [1,2,9]). Now, i.m n→∞ y n (φ 1 (s)))ds, Applying stochastic Lebesgue dominated convergence Theorem the operator F : Q → Q is continuous.

Uniqueness Theorem
Replace the assumptions (A1) and (A2) by (A * 1) and (A * 2), respectively, such that for all x ∈ L 2 (Ω) and satisfy the Lipschitz condition with respect to the second argument for all x ∈ L 2 (Ω) and satisfy the Lipschitz condition with respect to the second argument

Remark 1.
Let the assumptions (A * 1) and (A * 2) be satisfied, then we can obtain This implies that then (x 1 , y 1 ) = (x 2 , y 2 ) and the solution of the problem (1)-(3) is unique. Proof. Let (x,ŷ) be the solution of the coupled system

Continuous Dependence
which completes the proof. Proof. Let (x,ŷ) be the solutions of the coupled system Similarly we can obtain This implies that which completes the proof.

Solutions of the Problem (1), (2) and (4)
Define the mapping L(x, y) = (L 1 x, L 2 y) where L 1 x, L 2 y are given by the following stochastic integral equations Proof. Let x, y ∈ Q, , then we obtain and This implies that Proof. Let x, y ∈ Q, t 1 , t 2 ∈ [0, T] such that |t 2 − t 1 | < δ, then and However, then from (19) and (20), we deduce the equicontinuity of the class {L(x, y)(t)} on Q.

Existence Theorem
Now, we have the following existence theorem Proof. Let {(x n , y n )} ∈ Q be such that (x n , y n ) → (x, y) w.p.1.
Similarly we can obtain This implies that which completes the proof.

Conclusions
Here, we proved the existence of solutions of a coupled system of random and stochastic nonlinear differential equations with coupled nonlocal random and stochastic nonlinear integral conditions. The sufficient conditions for the uniqueness of the solution have been given. The continuous dependence of the unique solution has been studied.